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"... A q-query Locally Decodable Code (LDC) encodes an n-bit message x as an N-bit code-word C(x), such that one can probabilistically recover any bit xi of the message by querying only q bits of the codeword C(x), even after some constant fraction of codeword bits has been corrupted. We give new const ..."

A q-query Locally Decodable Code (LDC) encodes an n-bit message x as an N-bit code-word C(x), such that one can probabilistically recover any bit xi of the message by querying only q bits of the codeword C(x), even after some constant fraction of codeword bits has been corrupted. We give new constructions of three query LDCs of vastly shorter length than that of previous constructions. Specifically, given any Mersenne prime p = 2t −1, we design three query LDCs of length N = exp(O(n1/t)), for every n. Based on the largest known Mersenne prime, this translates to a length of less than exp(O(n10−7)), compared to exp(O(n1/2)) in the previous constructions. It has often been conjectured that there are infinitely many Mersenne primes. Under this conjecture, our constructions yield three query locally decodable codes of length N = exp(nO ( 1log log n)) for infinitely many n. We also obtain analogous improvements for Private Information Retrieval (PIR) schemes. We give 3-server PIR schemes with communication complexity of O(n10−7) to access an n-bit database, compared to the previous best scheme with complexity O(n1/5.25). Assuming again that there are infinitely many Mersenne primes, we get 3-server PIR schemes of communication complexity n O ( 1log log n) for infinitely many n. Previous families of LDCs and PIR schemes were based on the properties of low-degree multi-variate polynomials over finite fields. Our constructions are completely different and are obtained by constructing a large number of vectors in a small dimensional vector space whose inner products are restricted to lie in an algebraically nice set.

...90s, most notably in [3, 24, 21]. Katz and Trevisan [15] were the first to provide a formal definition of LDCs and prove lower bounds on their length. Further work on locally decodable codes includes =-=[5, 8, 20, 6, 16, 26]-=-. The length of optimal 2-query LDCs was settled by Kerenidis and de Wolf in [16] and is exp(n). 1 The length of optimal 3-query LDCs is unknown. The best upper bound prior to our work was exp � n 1/2...

"... I study the class of problems efficiently solvable by a quantum computer, given the ability to “postselect” on the outcomes of measurements. I prove that this class coincides with a classical complexity class called PP, or Probabilistic Polynomial-Time. Using this result, I show that several simple ..."

I study the class of problems efficiently solvable by a quantum computer, given the ability to “postselect” on the outcomes of measurements. I prove that this class coincides with a classical complexity class called PP, or Probabilistic Polynomial-Time. Using this result, I show that several simple changes to the axioms of quantum mechanics would let us solve PP-complete problems efficiently. The result also implies, as an easy corollary, a celebrated theorem of Beigel, Reingold, and Spielman that PP is closed under intersection, as well as a generalization of that theorem due to Fortnow and Reingold. This illustrates that quantum computing can yield new and simpler proofs of major results about classical computation.

"... Error-correcting codes and related combinatorial constructs play an important role in several recent (and old) results in computational complexity theory. In this paper we survey results on locally-testable and locally-decodable error-correcting codes, and their applications to complexity theory ..."

Error-correcting codes and related combinatorial constructs play an important role in several recent (and old) results in computational complexity theory. In this paper we survey results on locally-testable and locally-decodable error-correcting codes, and their applications to complexity theory and to cryptography.

"... Locally Decodable Codes (LDC) allow one to decode any particular symbol of the input message by making a constant number of queries to a codeword, even if a constant fraction of the codeword is damaged. In a recent work [Yek08] Yekhanin constructs a log n log log n 3-query LDC with sub-exponential l ..."

Locally Decodable Codes (LDC) allow one to decode any particular symbol of the input message by making a constant number of queries to a codeword, even if a constant fraction of the codeword is damaged. In a recent work [Yek08] Yekhanin constructs a log n log log n 3-query LDC with sub-exponential length of size exp(exp(O ())). However, this construction requires a conjecture that there are infinitely many Mersenne primes. In this paper we give the first unconditional constant query LDC construction with subexponantial codeword length. In addition our construction reduces codeword length. We give construction of 3-query LDC with codeword length exp(exp(O ( √ log n log log n))). Our construction also could be extended to higher number of queries. We give a 2r-query LDC with length of exp(exp(O ( r √ log n(log log n) r−1))). 1

...0]. The Hadamard code is the most famous 2query locally decodable code of length 2n. For two queries LDC tight lower bounds of 2θ(n) were given for linear codes in [GKST02] and for arbitrary codes in =-=[KdW03]-=-. The Katz and Trevisan [KT00] establish lower bounds of Ω̃(n2) for LDC with 3 queries and Ω̃(n1+1/(dq/2e−1)) for any number of queries q, and [Woo07] gives a slightly improves this bound. For many ye...

"... We show that the N P-Complete language 3SAT has a PCP verifier that makes two queries to a proof of almost-linear size and achieves sub-constant probability of error o(1). The verifier performs only projection tests, meaning that the answer to the first query determines at most one accepting answer ..."

We show that the N P-Complete language 3SAT has a PCP verifier that makes two queries to a proof of almost-linear size and achieves sub-constant probability of error o(1). The verifier performs only projection tests, meaning that the answer to the first query determines at most one accepting answer to the second query. Previously, by the parallel repetition theorem, there were PCP Theorems with two-query projection tests, but only (arbitrarily small) constant error and polynomial size [29]. There were also PCP Theorems with sub-constant error and almost-linear size, but a constant number of queries that is larger than 2 [26]. As a corollary, we obtain a host of new results. In particular, our theorem improves many of the hardness of approximation results that are proved using the parallel repetition theorem. A partial list includes the following: 1. 3SAT cannot be efficiently approximated to within a factor of 7 8 + o(1), unless P = N P. This holds even under almost-linear reductions. Previously, the best known N P-hardness

...ssumption that there are infinitely many Mersenne primes) are slightly sub-exponential 2no(1) and obtain a local decoder that queries 3 bits [35]. For two queries, an exponential lower bound is known =-=[22]-=-. For more queries, a super-linear lower bound is known [21]. Motivated by this state of affairs, Ben-Sasson et al [8] relaxed the notion of LDCs as to enable succinct constructions. Their idea was to...

"... We solve a 20-year old problem posed by Yannakakis and prove that there exists no polynomial-size linear program (LP) whose associated polytope projects to the traveling salesman polytope, even if the LP is not required to be symmetric. Moreover, we prove that this holds also for the cut polytope an ..."

We solve a 20-year old problem posed by Yannakakis and prove that there exists no polynomial-size linear program (LP) whose associated polytope projects to the traveling salesman polytope, even if the LP is not required to be symmetric. Moreover, we prove that this holds also for the cut polytope and the stable set polytope. These results were discovered through a new connection that we make between one-way quantum communication protocols and semidefinite programming reformulations of LPs.

"... This paper provides the first explicit construction of extractors which are simultaneously optimal up to constant factors in both seed length and output length. More precisely, for every n, k, our extractor uses a random seed of length O(log n) to transform any random source on n bits with (min-)ent ..."

This paper provides the first explicit construction of extractors which are simultaneously optimal up to constant factors in both seed length and output length. More precisely, for every n, k, our extractor uses a random seed of length O(log n) to transform any random source on n bits with (min-)entropy k, into a distribution on (1 − α)k bits that is ɛ-close to uniform. Here α and ɛ can be taken to be any positive constants. (In fact, ɛ can be almost polynomially small). Our improvements are obtained via three new techniques, each of which may be of independent interest. The first is a general construction of mergers [22] from locally decodable error-correcting codes. The second introduces new condensers that have constant seed length (and retain a constant fraction of the min-entropy in the random source). The third is a way to augment the “win-win repeated condensing” paradigm of [17] with error reduction techniques like [15] so that the our constant seed-length condensers can be used without error accumulation.

Abstract. We give the first exponential separation between quantum and bounded-error randomized one-way communication complexity. Specifically, we define the Hidden Matching Problem HMn: Alice gets as input a string x ∈ {0, 1} n and Bob gets a perfect matching M on the n coordinates. Bob’s goal is to output a tuple 〈i, j, b 〉 such that the edge (i, j) belongs to the matching M and b = xi ⊕ xj. We prove that the quantum one-way communication complexity of HMn is O(log n), yet any randomized one-way protocol with bounded error must use Ω ( √ n) bits of communication. No asymptotic gap for one-way communication was previously known. Our bounds also hold in the model of Simultaneous Messages (SM) and hence we provide the first exponential separation between quantum SM and randomized SM with public coins. For a Boolean decision version of HMn, we show that the quantum one-way communication complexity remains O(log n) and that the 0-error randomized one-way communication complexity is Ω(n). We prove that any randomized linear one-way protocol with bounded error for this problem requires Ω ( 3 √ n log n) bits of communication. Key words. Communication complexity, quantum computation, separation, hidden matching AMS subject classifications. 68P30,68Q15,68Q17,81P68 1. Introduction. The

... xi ⊕ xj. This problem is new, and we believe that its definition plays the major role in obtaining our result. The inspiration comes from the work by Kerenidis and de Wolf on locally decodable codes =-=[11]-=-. Let us give the intuition why this problem is hard for classical communication complexity protocols. Suppose (to make the problem even easier) that Bob’s matching M is restricted to be one of n fixe...

"... In this work we study two, seemingly unrelated, notions. Locally Decodable Codes (LDCs) are codes that allow the recovery of each message bit from a constant number of entries of the codeword. Polynomial Identity Testing (PIT) is one of the fundamental problems of algebraic complexity: we are given ..."

In this work we study two, seemingly unrelated, notions. Locally Decodable Codes (LDCs) are codes that allow the recovery of each message bit from a constant number of entries of the codeword. Polynomial Identity Testing (PIT) is one of the fundamental problems of algebraic complexity: we are given a circuit computing a multivariate polynomial and we have to determine whether the polynomial is identically zero. We improve known results on locally decodable codes and on polynomial identity testing and show a relation between the two notions. In particular we obtain the following results: 1. We show that if E: F n ↦ → F m is a linear LDC with 2 queries then m = exp(Ω(n)). Previously this was only known for fields of size &lt;&lt; 2 n [GKST01]. 2. We show that from every depth 3 arithmetic circuit (ΣΠΣ circuit), C, with a bounded (constant) top fan-in that computes the zero polynomial, one can construct a locally decodeable code. More formally: Assume that C is minimal (no subset of the multiplication gates sums to zero) and simple (no linear function appears in all the multiplication gates). Denote by d the degree of the polynomial computed by C and by r the rank of the linear

... case of codes with two queries (q = 2). Exponential lower bounds were first proved for linear codes [GKST01, Oba02] and then, by techniques from quantum computation, for non-linear codes over GF (2) =-=[KdW03]-=-. The bound of Goldreich et al [GKST01] actually holds for linear LDCs with 2 queries over any finite field, namely that m is at least 2 Ω(n)−log(|F|) , where F is the underlined field. This result is...

"... Alice wants to query a database but she does not want the database to learn what she is querying. She can ask for the entire database. Can she get her query answered with less communication? One model of this problem is Private Information Retrieval, henceforth PIR. We survey results obtained about ..."

Alice wants to query a database but she does not want the database to learn what she is querying. She can ask for the entire database. Can she get her query answered with less communication? One model of this problem is Private Information Retrieval, henceforth PIR. We survey results obtained about the PIR model including partial answers to the following questions. (1) What if there are k non-communicating copies of the database but they are computationally unbounded? (2) What if there is only one copy of the database and it is computationally bounded? 1

...addition I will have an extended version of this paper, with more proofs added, at www.eccc.unitier. de/eccc/ in 2004. To limit the survey the following topics are omitted. 1. Locally Decodable Codes =-=[27, 43, 44]-=-. 2. PIR's that are allowed to make errors but with low probability [43, 44]. 3. Quantum PIR's [44]. 4. Attempts to make PIR practical [4, 45] in the real-real world. 5. The connection between current...